SLAC-PUB-1598 June 1975 (T/E)
COHERENT PARITY VIOLATION”
A Review of Optical Activity with Massless and Massive Particles.
Gabriel Karl
Department of Physics University of Guelph, Guelph, CANADA?
and
I. Stanford Linear Accelerator Center Stanford University, Stanford, California 94305, U. S.A.
ABSTRACT
Recent theoretical speculations about parity violating effects in the
forward scattering of massless and massive particles are reviewed at
an elementary level. These phenomena are analogous to optical activity,
whose history is also briefly reviewed. Order of magnitude estimates
for the rotatory power are presented, and the feasibility of experiments
with neutron beams is discussed.
(Submitted for publication. )
*Work supported in part by the U. S. Energy Research and Development Administration and the National Research Council of Canada.
-i-Permanent address.
-l-
CONTENTS
1. Prehistory: Optical Activity with Light.
2. Wave Particle Duality: Optical Activity with Neutrons?
3. What Does One Have to Compute?
4. Light Scattering from Twisted Molecules.
5. Light Scattering from Atoms.
6. Forward Neutron Scattering: Weak Interactions.
7. Forward Neutron Scattering: Twisted Molecules.
8. How Small a Rotation of the Neutron Polarization Can
9.
10.
11.
12.
Be Measured?
How to Detect the Neutrino Sea.
Summary.
Acknowledgements.
References.
-2-
This is intended to be an elementary review of ideas connected with parity
violation in forward, coherent scattering. The subject is over 150 years old.
It is not my intention to give a complete survey, but to introduce the reader
quickly to recent theoretical speculations. The recent interest in phenomena of
this kind is mainly due to advances in our knowledge about weak interactions.
The experimental confirmation of neutral current interactions at CERN and NAL
has renewed interest in other weak processes expected theoretically. These
“new” weak interactions can be observed either in coherent or noncoherent
phenomena; this review deals + with coherent processes.
This review is based on talks which were supposed to be accessible to a
wide audience. Thus order of magnitude estimates of various effects are
stressed. The main danger of such estimates, in advance of experimental
knowledge, is that one has overlooked a factor of zero which is also present.
The estimates quoted are within a factor of ten of what I believe to be the correct
answer. All formulae below should not be taken more seriously than that. The
other self-imposed limitation is to restrict the discussion to scattering from
non-crystaline media: atomic and molecular fluids. One word about units: I
use the system, widespread in high energy physics, in which h=c=l. When
estimating numerically a formula in the CGS system a sprinkling of h, c factors
will convert the inverse masses into Compton wavelengths of the corresponding
particles.
1. Prehistory: Optical Activity with Light
At the beginning of the last century (Arago 1811, Biot 1812) it was discovered
experimentally that some substances are optically active. This was at a time
of debate about the nature of light: particles or waves? The recognition of two
independent states of polarized light helped to settle the issue, for a century,
-3-
in favor of transversely polarized waves. Biot recognized that when travelling
through an optically active medium (labelled right handed sugar in Fig. 1) plane
polarized light emerges with its plane of polarization rotated by an angle with
respect to the plane of polarization of the incident wave. Clearly the medium
can tell the difference betweenright and left. In modern language this is a
parity violating effect, and I believe this accounts for some of the original
excitement.
It can be seen from Fig. la that a plane polarized wave is not an eigenstate :.
of propagation through an optically active medium. The plane wave rotates into
another plane wave. The eigenstates of propagation through an active medium
are helicity states. This was recognized by Fresnel, who discovered helicity
states for light (Fresnel 1817, see also Jacob and Wick 1959) so called right
handed and left handed circularly polarized light waves. Fresnel also recognized
that right handed and left handed waves propagate at slightly different speeds
through active media, and this difference accounts for the rotation of polariza-
tion of plane waves (Fresnel 1823).
An important conceptual clarification was the experimental demonstration
(Pasteur 1848) that optically active substances come in two species: a right
turning variety and a left turning variety. Apart from their optical properties
the two species are identical in their physical and chemical behavior. The
crystals of the two species are in the relation of mirror images to each other.
The mirror image is not superimposable on the original by rotation, and Pasteur
and others, speculated correctly that the same relationship holds for the mole-
cules of the two optically active species. From a modern viewpoint Pasteur’s
experiments demonstrated that it is not the interaction (of the light wave with the
molecules) which violates parity but the target molecules themselves. The
-4-
optically active molecules are not in an eigenstate of parity. The interaction is
parity conserving. Indeed, the belief in parity conservation for all physical
interactions remained unquestioned for over a century (Lee and Yang 1956).
The fact that some molecules are quasistationary .in states which are not invari-
ant under parity is interesting in itself. This phenomenon is similar to spon-
taneous symmetry breaking in field theory. The hamiltonian of a sugar mole-
cule is parity conserving yet right handed sugar is not an eigenstate of parity.
2. Wave Particle Duality: Optical Activity with Neutrons? (Kabir 1971, :.
Podgoretskii 1966)
By the beginning of
pictures for light began
this century the duality between the wave and particle
to emerge. Light can be thought of as being composed
of massless particles travelling at high speed. In this picture the polarization
of the wave corresponds to the spin of the particle. In the particle picture the
optical rotation experiment looks, with a little license, as shown in Fig. lb.
Since the wave particle duality extends also to massive particles like electrons
or neutrons, one is tempted to ask: what happens if we replace the polarized
photon with a polarized massive particles? Kabir conjectured that polarized
neutrons would rotate their polarization when travelling through an optically
active medium. Podgoretskii had asked the same question somewhat earlier.
Of course, if the polarization of the neutron does rotate the next question is:
how many miles of sugar are needed for a one degree rotation? Neutrons of
course can also interact weakly without conserving parity and this can lead to a
rotation of the polarization by scattering from atoms rather than molecules
(Michel 1964, Stodolsky 1974). In what follows we shall discuss the order of
magnitude of these various phenomena.
-5 -
3. What Does One Have to Compute?
When travelling through a medium composed of scatterers the propagation
of a wave can be described by an index of refraction. The index of refraction is
related to the forward scattering amplitude f(0) which describes the scattering
of the wave by the atoms or molecules of the medium. The scattering amplitude
f(0) has the dimensions of length (the cross section is proportional to if(e) I 2).
In an optically active medium the index of refraction of right handed waves is
different from the index of refraction of left handed waves (Fresnel 1823), and f.
therefore the forward scattering amplitude of right handed waves fR(0) differs
from the forward scattering amplitude of left handed waves fL(0). In terms of
these quantities, the rotatory power (in radians per centimeter) of a medium
composed of N scatterers per cubic centimeter, for waves of wavelength A, is
given by*
t@ = N*h*Re FL(O) - k(O,]
Formula (1) is a transcription of Fresnel’s relation between the optical rotatory
power + and the indices of refraction nR, L n for right handed (R) and left handed
(L) waves. As stated in formula (1) it is the real part of the forward scattering
amplitude which governs the phase of the wave and therefore the phase mismatch
between R and L waves which accounts for the rotation a. If RefL(0) differs
from RefR(0) then, in general the imaginary parts are going to be different from
each other as well. The imaginary part of the forward scattering amplitude
governs the attenuation of the wave, when passing through the medium.
If Imf,(O) # ImfR(0) one of the two components R, L will get more attenuated
than the other one. In optics this effect is well known (Cotton 1895) under the
name of circular dichroism. The formula analogous to (l), giving the net
*For photons Eq. (1) has an extra factor of 2 -1 on the R. H. S.
-6-
longitudinal polarization q per centimeter, is for small 9!
+ = N- ha Im EL(O) - fR(0)l . (2)
In order to compute 9 or @ one has to find the spin dependent part of the forward
scattering amplitude fS = fL(0) -b(O).
In quantum mechanics the scattering amplitude fR LZis a matrix element ,
of an operator between the initial and final state. In forward scattering the final
state is identical to the initial state, and
: fR(0) = <“L AleI; A> (3)
where by A we represent all quantum numbers describing the state of the target
and of the beam apart from the helicity of the beam. If fR(0) # fL(0) then either
a) the state of the target is not an eigenstate of parity P IA> # f IA> or b) the
interaction operator 6 is not invariant under parity: PbP -l# 6. Both these
possibilities can be realized when scatterings neutrons from a medium as we
shall discuss. With light, unless electromagnetic interactions as we know them
are incorrect, only the first alternative holds.
It is amusing to note that it is possible to construct examples of states which
are not invariant under parity which nonetheless do not rotate polarized (neutron
or light) waves, after averaging over all orientations of the incident beam. In
other words, while the occurrence of spin rotation for isotropic media signals
a violation of parity the converse is not necessarily true.
4. Light Scattering from Twisted Molecules
To warm up we estimate the rotatory power of a medium composed of
twisted molecules. Light is scattered by the electrons in these molecules. The
scattering amplitude of light by an electron is the so-called classical radius of
-7-
the electron: e2/mc2. This is the only length which one can construct from
the classical quantities e, m, c. To obtain a difference between the fL amplitude
and fR scattering amplitude one must absorb the photon by electric dipole radia- -4
tion and re-emit by magnetic dipole radiation (or vice-versa), which have
opposite parity. Magnetic dipole amplitudes are smaller than electric dipole
amplitudes by a factor ka- a where a is the molecular size and h the wave-
length of the radiation, so that:
‘. e2 fS = fL(0) - fR(0) = 7 . ka mc
and
Cp = N.h.fS -N-A e2 s2.kazNs mc
(5)
In the last step, I took as typical molecular size a = li2/me2 which brought in
Planck’s constant. For typical densities N - 1021 molecules/cm3, Eq. (5) gives
ib - 1 rad/cm. This is the correct order of magnitude, which is known experi-
mentally. The dependence of + on the wavelength, the “dispersion” of the
rotatory power is not given in Eq. (5). As the wavelength h increases relative
to molecular size the rotary power 9 has to vanish (Boltzmann 1874). The
theory of optical activity is an elaborate field of study in molecular physics; the
general scale of rotatory power is correctly given in Eq. (5).
5. Light Scattering from Atoms
The discovery of parity violation in weak interactions started speculations
about optical activity with atoms. A priori , there could exist a weak, parity
violating interaction between electrons and protons (Zelldovich 1959). So
far there is no evidence for this interaction. This possibility gained support
after the discovery of neutral currents: hadronic transitions induced by
-8 -
neutrinos unaccompanied by muons. Theories which predict neutral currents
also predict a weak electron-nucleon interaction (see for example Weinberg
1967 and Salam 1968 which are leptonic models of such theories). If such an
interaction exists then, in an atom it would give rise to a small mixing of nega-
tive parity states into positive parity states and vice versa (Zel’dovich 1959).
The existence of such a mixing would be revealed in parity violating experiments,
in particular in forward, coherent scattering of light (Zel’dovich 1959) from
hydrogen atoms. More recently, Khriplovich emphasized the advantage of :.
looking at heavy atoms, and gave an estimate for the size of the effect in TI
vapor (Khriplovich 19 74). With angular momentum conservation, the parity
violating interaction between an electron in an atom and the nucleus of n
nucleons is of the form (Bouchiat and Bouchiat 1974):
(6)
where G is the Fermi weak coupling constant and m,F, iz are the electron
mass, momentum and spin. With this interaction the (amplitude of) admixture
of the wrong parity component P l/2
into the state S1,2 of a hydrogenic atom of
nuclear charge Z has the order of magnitude:
Wrong parity amplitude - (Gm2) a2Z2n (7)
and therefore one expects a rotatory power:
For a gas of heavy atoms Z N 80,~ N 200 at 10 19 atoms/cm3 at a wavelength of
1O-4 cm formula (8) gives + N 10m8cm -1 which is within a factor of ten of the
value quoted by Khriplovich, who takes into account relativistic corrections not
mentioned here. Although the expected effect is small it is measurable, and
-9-
experiments are in preparation to detect this rotation (Sandars 1975). A non-
zero effect would demonstrate either the existence of a parity violating electron-
nucleus interaction, or a direct parity violation in the interaction of light with
matter. There is no need to emphasize the importance of such information.
6. Forward Neutron Scattering: Weak Interactions
The neutron participates in weak interactions, which are parity violating.
One might thus expect that in forward neutron scattering through matter, there
is a weak, parity violating spin dependence. The existence of such an effect :.
does not hinge on the existence of neutral currents: the weak, charge changing
process:
w - pn is sufficient to produce such a parity violation (Michel 1964, Stodolsky 19 74).
In the case at hand it is clearly the interaction which is parity violating
rather than the initial target state. An order of magnitude of the expected effect
may be constructed using the scale set by the weak interaction coupling constant
G: --
A
fS N GmnG ZA n
(9)
where ZA is the axial vector charge of the nucleus, essentially the number of
l- protons and mn, F and 2 (T are the neutron mass, momentum and spin. The
rotatory power of matter composed of heavy atoms at normal densities is:
a-N hfS-NGZA N 10-8rad/cm . (10)
where I have taken N - 10 22 at/cm3 and ZA - 100. The estimate (10) is identical
to those published in the literature (Michel 1964, Stodolsky 1974). Stodolsky
also stated the associated effect in absorption, and the possibility of deducing
- 10 -
the sign of G from the sign of Cp. This could be very useful to confirm or rule
out theories for weak interactions based on intermediate bosons in which the
sign of G is expected to be positive.
7. Forward Neutron Scattering from Twisted Molecules
The parity violation in forward neutron scattering can also occur if the
target is not in an eigenstate of parity-e. g. , the nucleus of an “activel* atom
in a handed molecule. To understand this it is useful to discuss a simple
model (Kabir et al. 1974), of a single spinless nucleus of mass M bound in a --
parity violating (anisotropic) harmonic oscillator well:
-2
H=2M 3 p+ lM + Qxyz (11) c
invariant under parity odd under parity
A similar model has been proposed much earlier (cf. Condon 1937) to describe
the motion of an electron in an optically active molecule. Here, the hamiltonian
(11) describes the zero point motion of a nucleus, in such a molecule. To
visualize this motion it is convenient to think of the nucleus moving on a handed
spiral, as in Fig. 2. How does the spin of the forward scattered neutron find
the handedness of the orbital motion of the nucleus? A possible mechanism is
shown in Fig. 2. This is a resonant mechanism: the incident polarized neutron
is absorbed in the nucleus and carried along for a while. Since the nucleus is
initially spinless, the spin of the neutron becomes the spin of the intermediate
nuclear resonance. Due to spin-orbit coupling the resonance rotates its spin
and, when the neutron is re-emitted its spin has been turned a little. This
mechanism is “operative” when the lifetime of the nuclear resonance is compa-
rable to the period of oscillation of the nucleus on the “spiral.” The mechanism
- 11 -
giving rise to spin orbit coupling in the second stage (on Fig. 2) is the coupling
of the spin magnetic moment of the resonance to the magnetic field generated
by the orbital motion of the nucleus. The resulting spin dependence of the
forward scattering amplitude is easy to estimate: T
‘spin’ Porbit
fS=fL -fR-fnucl * r3 - tw
lilw (12)
where the spin magnetic moment of the resonance ,u spin is roughly (eh/mnc), L.
the orbital magnetic moment porbit - (Z eh/Mc) in an orbit of size r -fi
and the factor kr accounts for the p-wave emission (or absorption) of the
neutron. The scale of the forward amplitude fS is set by the scattering ampli-
tude from a free nucleus fnucl. Replacing all these factors in Eq. (12) we
obtain:
fL - fR = fnucl l Z . k
and therefore
9 = N. A- (fL-fR) = Na fnucl* Z
(13)
(14)
where I took fnucl - 10 -13 cm, Z - 100 and N - 1022cm-3 for a heavy nucleus,
with a fairly large neutron scattering amplitude.
In the earliest treatment of this problem (Baryshevskii 1966) the tacit
assumption of a rigid molecule was made. With this assumption parity violation
can only occur due to the multiple scattering of the neutron from different nuclei
in the molecule. This greatly reduces the size of the effect (see also Kabir
et al. 1975). --
I
- 12 -
8. How Small a Rotation of Neutron Polarization Can Be Detected?
The most accurate experiments which detect the rotation of the polariza-
tion of a neutron beam are Ramsey’s well known experiments to measure the
elusive electric dipole moment of the neutron (e. g., Miller et al. 1967). The --
rotation of the neutron polarization occurs because of the electric field through
which the neutrons travel. The present limit for the electric dipole moment
of the neutron is about 10 -24 cm (Ramsey 1975), which for an electric field of
lo4 volts/cm corresponds to an energy difference of 10e2’ eV. The corre-
sponding precession period is lo4 sec. In other words, if a neutron had an
electric dipole moment of 10 -24 cm, it would make a full precession in 10 hours,
in the electric field mentioned above. Of course the neutrons don’t hang on
that long in the electric field. The measurement time in Ramsey’s apparatus
can be obtained by dividing the length of the apparatus (- 1 m) by the speed of
neutrons (lo2 m/set). During a time of 10 -2 set the neutrons rotate their spin
polarization by an angle of 10 -6 radians. Thus it is possible, at the present
time to measure a neutron angle of precession of 10 -6 radians. (See also
Mezei 1972. ) This angle is
Roughly speaking one would
by a piece of matter.
in the range of the effects discussed in this review.
have to replace the electric field in the experiment
9. How to Detect the Neutrino Sea
According to rumors originating in the astrophysics community we are all
immersed in a neutrino (or antineutrino) sea. The sea is very hard to detect
experimentally and the corresponding cosmological theories hard to disprove.
Can one detect the neutrino sea by parity violating coherent experiments?
Drell asked this question from Royer who tackled it (Royer 1968). Royer found
that a polarized plane wave of light will rotate its polarization slowly when
travelling through the neutrino sea. The density of the neutrino sea is
- 13 -
characterized by its Fermi energy kF. We construct an inverse length to
compute the rotatory power 4, of the Fermi sea, using the weak interaction
coupling constant G and kF:*
4,- Gk$ N lo-l4 rad/cm . ’ (15)
I have inserted a factor Q! to account for the dissociation of the photon into a
lepton pair prior to interacting with the neutrinos, and the subsequent recom-
bination of the lepton pair. In evaluating (15) I have assumed kF - 100 eV.
The estimate (15) agrees with Royer, apart from a numerical factor (9r4) which
is hard to get by waving one’s hands. Note that a rotatory power as small as
lo-l4 rad/cm is perhaps not impossible to detect. A plane polarized laser
beam, starting on the Moon would be rotated on Earth by 10 -4 radians. The
orientation of the initial plane of polarization could be referred to the position
of stars. It is not obvious how much an estimate like (15) would be changed by
taking neutral currents into account. In any case the experiment might well be
worth performing just in case of the neutrino sea is really here and is much
deeper than expected.
The analogous observation of the neutrino sea with massive polarized
particles has also been proposed (Stodolsky 1975). Stodolsky pointed out that a
transversely polarized electron would slowly rotate its polarization while
travelling through the neutrino sea. The estimate of the rotatory power of the
neutrino sea for electrons is similar to (15). The factor of Q! is no longer
present since we have a lepton to start with. If the electron sails through the
neutrino sea by virtue of the motion of the solar system around the galactic
center (v/c - 10w3) the attendant spin rotation is of the order 1000 radians/year
*It is amusing to note that both these concepts are due to Fermi.
- 14 -
for a neutrino sea of % = 100 eV:
Gg . 3. IO7 s x 3. lOlo 2 x 10v3 N lo3 rad/year .
Stodolsky’s estimate is similar to this one. A method of this kind, involving
massive particles might be sensitive to much shallower neutrino seas than
% - 100 eV.
10. Summary
In addition to the well known coherent parity violation in the propagation of
light through handed media a number of similar new phenomena have been
proposed. These are all summarized in Table I. Experimental information
on these effects is eagerly awaited.
11. Acknowledgements
I am indebted to P. K. Kabir for many things: in particular for involving
me into this subject. I also thank S. Drell, P. Egelstaff, T. Ericson, C. Gray,
E. Obryk, J. D. Poll, N. Ramsey, L. Stodolsky, J. Van Kranendonk, and
L. Wolfenstein for conversations which I found helpful. Finally I am indebted
to my wife and the Theory Group at SLAC for the opportunity to visit Stanford
University where this review was written.
I
- 15 -
12. References
Arago D. G. 1811, Mem. Inst. de France p. 93-134.
Baryshevskii V. G., 1966, Iadernaia Fizika 4, 72.
Biot J. B. , 1812, Mem. Inst. de France p. l-372.
Boltzmann L. , 1874, Pogg. Ann. Phys. Chem. p. ‘128.
Bouchiat M. A. and Bouchiat C. C. , 1974, Phys . Lett. e, 111.
Condon E. , 1937, Rev. Mod. Phys. 2, 432.
Cotton A. , 1895, Compt. Rend. 120, 989.
Fresnel A., 1817, Mem. Inst. de France, Nov. 10.
Fresnel A., 1823, Mem. Acad. de Sci. Jan. 7.
Jacob M. and Wick G. C., 1959, Ann. Phys. (N. Y.) 2, 404.
Kabir P. K., 1971, private communication.
Kabir P. K. , Karl G. and Obryk E. , 1974, Phys. Rev. DlO, 1471.
Kabir P. K. , Karl G. and Obryk E. , 1975, to be published.
Khriplovich I. B. , 1974, J.E.T.P. Letters 20, 686. I am indebted to
S. Brodsky for this reference.
Lee T. D. and Yang C. N., 1956, Phys. Rev. 2, 254.
Mezei F. 1972, Zeits f. Physik. 255, 146.
Michel F. Curtis, 1964, Phys. Rev. 133B, 329.
Miller P. D., Dress W. B., Baird J. K. and Ramsey N. F., 1967, Phys.
Rev. Lett. 2, 381.
Pasteur L. , 1848, Ann. de Chimie et de Physique 24, 442.
Podgoretskii I. M., 1966, see the acknowledgement of Baryshevskii, 1966.
Ramsey N. F. 1975, private communication.
Royer J., 1968, Phys. Rev. 174, 1719.
Sandars P. G. H., 1975, Toronto lecture.
Salam A. , 1968, in Elementary Particle Physics, edited by N. Svartholm
- 16 -
Stodolsky L. , 1974, Phys. Letters s, 352.
Stodolsky L. , 1975, Phys. Rev. Letters 34, 110.
Weinberg S. , 1967, Rev. Letters Phys. 2, 1264.
Zel’dovich Ya. B. , 1959, J.E.T.P. 36, 964. 1
- 1; -
TABLE I
Coherent Parity Violating Effects
Effect Rotatory Power
a) Light Propagation:
1. States of mixed parity in molecules
2. Statescf mixed parity in atoms (due to neutral currents)
1 rad/cm
10q7 rad/cm
b) Neutron Propagation:
1. States of mixed parity in molecules 1o-5 rad/cm
2. Parity violating weak interactions with nuclei 1O-8 rad/cm (charged and neutral currents)
- 18 -
Figure Captions
1. Wave and particle pictures of optical activity.
2. The resonant mechanism for neutron optical activity.
I
(a) Wove Picture
Incident -L- Outgoing Light Wave Target Light Wave
) (.b)
Particle Picture
Incident Photon
-L- Outgoing Target Photon
QUESTION (Kabir, Podgoretskii) (0) Is there an analogous effect with massive particles, say neutrons ? (b) If there is; how many miles L of “Right-Handed Sugar” do we need
for a one degree rotation of the polarization? 1711.1
Fig. 1
Incident Neutron
Handed Outgoing Target Neutron
I
Neutron Absorption
II
Spin Rotation
Neutron Emission
14: L 2728A2
Fig.. 2